Diamond

Diamond is one of the most famous allotropes of carbon, because its hardness and high dispersion of light means that it looks shiny, and is very resistant to wear. Thus, it is used for jewellery and for cutting through extremely hard things (industrial use).

In the diamond lattice, each carbon atom is covalently bonded to four other carbons in a tetrahedron. Thus, when joined together, these form rings of carbon with a hexagonal shape, similar to the shape of the glucose molecule :).

Graphite

Graphite is one of the most common allotropes of carbon. It has many many functions in our daily life, due to its shape. In Graphite, each carbon molecule only uses 3 of its 4 valence electrons to bond to other carbon atoms in a trigonal planar shape. Thus, there are free electrons, which allows it to conduct electricity. Also, this means that graphite is in sheets, which means that you can use it in a pencil, to layer down sheets of carbon atoms to leave marks on the paper. It can also be used as a lubricant.

C60 Fullerene

This is the kind of carbon you find in soot. The arrangement of carbon atoms in this molecule is quite similar to graphite, as although it is spherical, each carbon atom is bonded to three other carbon atoms in a trigonal planar shape. Thus, it is also sort of in a layer. There are also thus free electrons, which means that it can conduct electricity. The spheres can move around relative to each other, and it is thus a lubricant, though a lot less effective than graphite.